In order to address the limitations of linear, Gaussian methods for sensor placement and feature selection demonstrated in the first half of the paper, we develop a novel data-driven approach based on consideration of secant vectors between states in Section .
Related secant-based approaches have been pioneered by [1]}, [2]}, [3]}, [4]} for the purpose of finding optimal embedding subspaces.
While their considerations of secants lead to continuous optimization problems over subspaces, our considerations of secants lead to combinatorial optimization problems over sets of sensors.
We develop three different secant-based objectives together with greedy algorithms that each provide different types of robust, near-minimal reconstruction guarantees for very general types of nonlinear inverse problems.
The guarantees stem from the underlying geometric information that is captured by secants and encoded in our optimization objectives.
Moreover, the objectives we consider each have the celebrated diminishing returns property called submodularity, allowing us to leverage the classical results of G. L. Nemhauser and L. A. Wolsey et al. [5]}, [6]} to guarantee the performance of efficient greedy algorithms for sensor placement.
We also leverage concentration of measure results in order to prove performance guarantees when the secants are randomly down-sampled, enabling computational scalability to very large data sets.
Each of these techniques demonstrates greatly improved performance compared to a large collection of linear techniques on a canonical shock-mixing layer flow problem [7]} as well as for selecting fundamental manifold learning coordinates.
